Preprint typeset in JHEP style. - HYPER VERSION SCIPP-01/10

hep-ph/yymmnnn

Can the Higgs sector contribute significantly to the muon anomalous magnetic moment?

Athanasios Dedes

Physikalisches Institut der Universität Bonn,

Nußallee 12, D-53115 Bonn, Germany

  E-mail: dedes@th.physik.uni-bonn.de

Howard E. Haber

Santa Cruz Institute for Particle Physics

University of California, Santa Cruz, CA, 95064 USA

  E-mail: haber@scipp.ucsc.edu

Abstract: A light CP-even Higgs boson with m_{h} ∼ 10 GeV could explain the recent BNL measurement of the muon anomalous magnetic moment, in the framework of a general two-Higgs-doublet extension of the Standard Model. However, the allowed Higgs mass window is quite small and the corresponding model parameters are very constrained. The Higgs sector can contribute significantly to the observed BNL result for g − 2 without violating known experimental constraints only if the hZZ coupling (approximately) vanishes and M_{Υ} < ∼ m_{h} < ∼ 2m_{B}.

Keywords: Beyond Standard Model.

Contents

1. Introduction 1

2. Higgs boson corrections to the muon anomalous magnetic moment 5 2.1 Non-decoupling limit: sin(β − α) = 0 6 2.2 Decoupling limit: cos(β − α) = 0 7

3. CESR and LEP constraints on the light Higgs boson 8

4. Final Results and Conclusions 10

1. Introduction

A new experimental value of the muon anomalous magnetic moment, a_{μ} ≡ \frac{1}{2}(g −2)_{μ}, measured at BNL, was recently reported in ref. [1]. Comparing the measured value to its predicted value in the Standard Model (SM), ref. [1] reported that

a_{μ}^{exp} − a_{μ}^{SM} = 426 ± 165 × 10^{−11} .  (1.1)

Ref. [2] has reviewed the Standard Model computation of a_{μ} and concluded that if the deviation of eq. (1.1) can be attributed to new physics effects [δa_{μ}^{NP}], then at 90% CL, δa_{μ}^{NP} must lie in the range

215 × 10^{−11} < ∼ δa_{μ}^{NP} < ∼ 637 × 10^{−11} .  (1.2)

This contribution is positive, and is of the order of the electroweak corrections to a_{μ}. More precisely, the contribution needed from new physics effects has to be of the order of G_{F} m_{μ}^{2}/(4π^{2}\sqrt{2}), where G_{F} = 1.16637(1) × 10^{−5} GeV^{−2} is Fermi’s constant. In this paper, we consider the possibility that δa_{μ}^{NP} arises entirely from the Higgs sector. In the SM, the Higgs boson contribution to a_{μ} is further suppressed (relative to the main electroweak contribution) by a factor of m_{μ}^{2}/m_{h}^{2}. In light of the recent SM Higgs mass limit, m_{h} > ∼ 113.5 GeV obtained at the LEP collider [3], the SM Higgs contribution to a_{μ} is clearly negligible.

However, the Higgs sector contribution to a_{μ} could be considerably enhanced in a two-Higgs-doublet extension of the Standard Model (2HDM). The significance of the (g −2)_{μ} constraint for the 2HDM (in light of the LEP Higgs constraints) was emphasized in ref. [4], where the constraints of the previous BNL (g −2)_{μ} measurements were analyzed and the implications of future (g −2)_{μ} measurements were considered. Now that we have the first possible indication of δa_{μ}^{NP} ≠ 0, it is appropriate to revisit the question of the Higgs sector contribution to a_{μ}.

The enhancement of the Higgs sector contribution to a_{μ} relative to the SM result can arise from two different effects. First, an enhanced hμ^{+}μ^{−} coupling proportional to the ratio of Higgs vacuum expectation values, tan β, yields a Higgs contribution to δa_{μ}^{NP} proportional to tan^{2} β. Second, a suppressed hZZ coupling, proportional to sin(β − α) [using notation reviewed below], can permit the existence of a CP-even Higgs boson mass substantially below the LEP SM Higgs mass limit. In units of G_{F} m_{μ}^{2}/(4π^{2}\sqrt{2}), the overall enhancement is of order

\frac{m_{μ}^{2}}{m_{h}^{2}} × tan^{2} β × F (\frac{m_{μ}^{2}}{m_{h}^{2}} ) ≃ 1—10 .  (1.3)

F (x) is a loop factor which involves logarithms of the form ln(m_{h}^{2}/m_{μ}^{2}) ∼ O(10). A light CP-even Higgs boson with m_{h} ≃ 10 GeV and 30 < ∼ tan β < ∼ 50, predicts a muon anomalous magnetic moment to lie in the 90% CL allowed range for new physics effects specified in eq. (1.2).

A 2HDM in which the Higgs sector contribution to δa_{μ}^{NP} is significant is not compatible with the Higgs sector of the minimal supersymmetric extension of the Standard Model (MSSM). This is true because one cannot have a very light h with suppressed hZZ couplings without an observable rate for Z → hA, in conflict with LEP data [5]. Moreover, the MSSM provides additional mechanisms for generating significant contributions to δa_{μ}^{NP}. A number of recent papers [6, 7, 8, 9, 10] have shown that the recent BNL measurement is compatible with supersymmetric contributions to δa_{μ}^{NP} involving chargino and neutralino exchange, over an interesting region of MSSM parameter space.

In this paper, we focus on the possibility that the new physics contribution to a_{μ} arises solely from the Higgs sector. The two-doublet Higgs sector [11] contains eight scalar degrees of freedom. It is convenient to distinguish between the two doublets by employing one complex Y = −1 doublet, Φ_{d}= (Φ_{d}^{0}, Φ_{d}^{−}) and one complex Y = +1 doublet, Φ_{u}= (Φ_{u}^{+}, Φ_{u}^{0}). To avoid Higgs-mediated flavor changing neutral currents, we do not allow the most general Higgs-fermion interaction [12]. Instead, we impose discrete symmetries (which may be softly-broken by mass terms), and consider two possible models [13]. In Model I, Φ_{d}^{0} couples to both up-type and down-type quark and lepton pairs, while the coupling of Φ_{u}^{0} to fermion pairs is absent.^{1} In Model II, Φ_{d}^{0} [Φ_{u}^{0}] couples exclusively to down-type [up-type] fermion pairs. When the Higgs potential is minimized, the neutral components of the Higgs fields acquire vacuum expectation values:^{2}

1One can just as well assume that Φ_{u}^{0} couples to both up-type and down-type quark and lepton pairs, while the coupling of Φ_{d}^{0} to fermion pairs is absent. In this case, the all the results of this paper would apply simply by replacing tan β with cot β.

⟨Φ_{d}⟩ = \frac{1}{\sqrt{2}} ( \begin{matrix}v_{d} \\ 0 \end{matrix} ) ,  ⟨Φ_{u}⟩ = \frac{1}{\sqrt{2}} ( \begin{matrix}0 \\ v_{u} \end{matrix} ) ,  (1.4)

where the normalization has been chosen such that v^{2} ≡ v_{d}^{2} + v_{u}^{2} = (246 GeV)^{2}, while the ratio tan β ≡ v_{u}/v_{d} is a free parameter of the model. The physical Higgs spectrum consists of a charged Higgs pair

H^{±} = Φ_{d}^{±} sin β + Φ_{u}^{±} cos β ,  (1.5)

one CP-odd scalar

A = \sqrt{2} (Im Φ_{d}^{0} sin β + Im Φ_{u}^{0} cos β) ,  (1.6)

and two CP-even scalars:

h = −(\sqrt{2} Re Φ_{d}^{0} − v_{d}) sin α + (\sqrt{2} Re Φ_{u}^{0} − v_{u}) cos α ,
H = (\sqrt{2} Re Φ_{d}^{0} − v_{d}) cos α + (\sqrt{2} Re Φ_{u}^{0} − v_{u}) sin α ,  (1.7)

(with m_{h} ≤ m_{H}). The angle α arises when the CP-even Higgs squared-mass matrix (in the Φ_{d}^{0}—Φ_{u}^{0} basis) is diagonalized to obtain the physical CP-even Higgs states.

We briefly review the Higgs couplings relevant for our analysis. The tree-level h couplings to ZZ and AZ are given by

g_{hZZ} = \frac{gm_{Z} sin(β − α)}{cos θ_{W}} ,  (1.8)
g_{hAZ} = \frac{g cos(β − α)}{2 cos θ_{W}} .  (1.9)

For the corresponding couplings of H to ZZ and AZ, one must interchange sin(β−α) and cos(β − α) in the above formulae.

The pattern of couplings of the Higgs bosons to fermions depends on the choice of model. However, in this paper we are mainly concerned with the coupling of down-type fermions to Higgs bosons, which are the same in Model I and Model II. For our analysis, the relevant couplings of the neutral Higgs bosons to bb̄ or μ^{+}μ^{−} relative to the SM value, m_{f}/v [f= b or μ], are given by

hbb̄ (or hμ^{+}μ^{−}) : − \frac{sin α}{cos β} = sin(β − α) − tan β cos(β − α) ,  (1.10)
Hbb̄ (or Hμ^{+}μ^{−}) : \frac{cos α}{cos β} = cos(β − α) + tan β sin(β − α) ,  (1.11)
Abb̄ (or Aμ^{+}μ^{−}) : γ_{5} tan β ,  (1.12)

2In this paper, we neglect the possibility of significant CP-violation in the Higgs sector. In this case, the phases of the Higgs fields can be chosen such that the vacuum expectation values are real and positive.

(the γ_{5} indicates a pseudoscalar coupling), and the charged Higgs boson couplings to muon pairs (with all particles pointing into the vertex) is given by

g_{H^{−}μ^{+}ν} = \frac{gm_{μ}}{\sqrt{2}m_{W}} tan β P_{L} ,  (1.13)

where P_{L} ≡ \frac{1}{2}(1 − γ_{5}).

We have noted above that only light Higgs bosons with enhanced couplings to down-type fermions can contribute appreciably to δa_{μ}^{NP}. To avoid the LEP SM Higgs mass limit, such a light Higgs boson should be almost decoupled from ZZ. This implies that either h is light, with | sin(β − α)| ≪ 1 [see eq. (1.8)] or A is light (since A has no tree-level coupling to vector boson pairs). In the next section, we will show that a light A makes a negative contribution to δa_{μ}^{NP} and thus is not compatible with the recent BNL measurement. Hence, we focus on the 2HDM in which only h is light and sin(β − α) ≃ 0. From eq. (1.10), we see that if sin(β − α) ≃ 0, then the coupling of h to down-type fermions is proportional to tan β. Thus, in the region of large tan β and small sin(β − α), the contribution of a light CP-even Higgs boson of the 2HDM may yield a significant correction to δa_{μ}^{NP} without being in conflict with the LEP SM Higgs search.

Although the considerations above apply to both Model I and Model II, it is important to note that the Higgs couplings to up-type fermions differ between the two models. For example, in Model II the coupling of h to tt̄ relative to the SM value, m_{t}/v, is given by:

htt̄ (or hμ^{+}μ^{−}) : \frac{cos α}{sin β} = sin(β − α) + cot β cos(β − α) ,  (1.14)

which in the limit of sin(β−α) = 0 reduces to cot β; i.e., the htt̄coupling is suppressed in Model II for large tan β. In contrast, the Model I htt̄ coupling relative to the SM value is the same as in Model II. That is, for sin(β − α) = 0, this coupling is |g_{htt̄}| = (m_{t}/v) tan β ≫ 1 for large tan β. Thus, the tan β enhanced Higgs couplings to tt̄ are non-perturbative. Both theoretical and experimental considerations lead us to reject this possibility. Henceforth, we will assume that the 2HDM contains Model II Higgs-fermion couplings.

Finally, we note that in the parameter region cited above, the heavier Higgs bosons, H, A, H^{±}, cannot be arbitrarily heavy. If one attempts to take such a limit, one finds that there must be some Higgs quartic self-couplings that become significantly larger than 1 [14]. That is, this model does not possess a decoupling limit. However, the model stays weakly coupled as long as the heavier Higgs states are not too much larger than v = 246 GeV. In contrast, in the limit of cos(β − α) = 0, the couplings of h reduce to those of the SM Higgs boson. This decoupling limit can be formally reached by taking the masses of H, A, H^{±} to be arbitrarily large, while keeping the quartic Higgs self-couplings < ∼ O(1) [14]. The resulting low-energy effective theory is just the SM with one Higgs doublet. Of course, as we have noted above, the contribution of SM Higgs boson to δa_{μ}^{NP} is negligible. Thus, over an intermediate range of heavy Higgs masses, the contribution of H, A, H^{±} (which are tan^{2} β enhanced) to δa_{μ}^{NP} will be significantly larger than that of h even though cos(β − α) ≃ 0.

2. Higgs boson corrections to the muon anomalous magnetic moment

The first calculation of the one-loop electroweak corrections to the muon anomalous magnetic moment was presented by Weinberg and Jackiw [15] and by Fujikawa, Lee and Sanda [16]. A very useful compendium of formaule for the one-loop corrections to g − 2 in a general electroweak model was given in ref. [17], and applied to the 2HDM in ref. [18].^{3} In the 2HDM, both neutral and charged Higgs bosons contribute to g − 2. A convenient list of the relevant formulae can be found in ref. [4].

δa_{μ}^{h} = \frac{G_{F} m_{μ}^{2}}{4π^{2}\sqrt{2}} (\frac{sin α}{cos β} )^{2} R_{h} F_{h}(R_{h})  (2.1)
δa_{μ}^{H} = \frac{G_{F} m_{μ}^{2}}{4π^{2}\sqrt{2}} (\frac{cos α}{cos β})^{2} R_{H} F_{H}(R_{H})  (2.2)
δa_{μ}^{A} = \frac{G_{F} m_{μ}^{2}}{4π^{2}\sqrt{2}} tan^{2} β R_{A} F_{A}(R_{A})  (2.3)
δa_{μ}^{H^{±}} = \frac{G_{F} m_{μ}^{2}}{4π^{2}\sqrt{2}} tan^{2} β R_{H^{±}} F_{H^{±}}(R_{H^{±}}, R_{ν})  (2.4)

where R_{h,H,A,H^{±}} ≡ m_{μ}^{2}/m_{h,H,A,H^{±}}^{2}, R_{ν} ≡ m_{ν}^{2}/m_{H^{±}}^{2} and

F_{h,H}(R_{h,H}) = ∫_{0}^{1} dx \frac{x^{2}(2 − x)}{R_{h,H}x^{2} − x + 1} ,  (2.5)
F_{A}(R_{A}) = ∫_{0}^{1} dx \frac{−x^{3}}{R_{A}x^{2} − x + 1} ,  (2.6)
F_{H^{±}}(R_{H^{±}}, R_{ν}) = ∫_{0}^{1} dx \frac{x^{2}(1 − x)}{R_{H^{±}}x^{2} + (1 − R_{H^{±}} − R_{ν})x + R_{ν}} ,  (2.7)

The neutrino mass is negligible, so henceforth we set R_{ν} = 0. Since R_{h,H,A,H^{±}} ≪ 1, one can easily expand the above integrals in the corresponding small parameter. In the next two subsections, we write out the leading terms in this expansion, which are quite accurate in the Higgs mass range of interest.^{4} 2.1 Non-decoupling limit: sin(β − α) = 0

3We correct a small error in the expression for the charged Higgs contribution given in ref. [18].

4The plot shown in this paper is based on the exact values of the above integrals.

In section 1, we argued that the most significant Higgs contribution to δa_{μ}^{NP} (consistent with the LEP SM Higgs search) arises in the parameter regime in which sin(β − α) ≃ 0 and tan β ≫ 1. Setting sin(β − α) = 0 and keeping only the leading terms in R when evaluating the above integrals, the total Higgs sector contribution to a_{μ} is given by:

δa_{μ}^{Higgs} = δa_{μ}^{h} + δa_{μ}^{H} + δa_{μ}^{A} + δa_{μ}^{H^{±}}

≃ \frac{G_{F} m_{μ}^{2}}{4π^{2}\sqrt{2}} tan^{2} β\{\frac{m_{μ}^{2}}{m_{h}^{2}} [ln (\frac{m_{h}^{2}}{m_{μ}^{2}} ) − \frac{7}{6}] − \frac{m_{μ}^{2}}{m_{A}^{2}} [ln (\frac{m_{A}^{2}}{m_{μ}^{2}} ) − \frac{11}{6}] − \frac{m_{μ}^{2}}{6m_{H^{±}}^{2}} \} .

(2.8)

Note that the logarithms appearing in eq. (2.8) always dominate the corresponding constant terms when the Higgs masses are larger than 1 GeV. It is then clear that A and H^{±} exchange contribute a negative value to δa_{μ}^{NP}. Since our goal is to explain the BNL g − 2 measurement which suggests a positive value for δa_{μ}^{NP}, we should take m_{A} and m_{H^{±}} large (masses above 100 GeV are sufficient) in order that the corresponding A and H^{±} negative contributions are neglibly small.^{5} If δa_{μ}^{NP} is to be a consequence of the Higgs sector, it must be entirely due to the contribution of the light CP-even Higgs boson. Note that the heavier CP-even Higgs, H, does not give a contribution proportional to tan β (as shown in section 1); hence its contribution to δa_{μ}^{NP} can be neglected in eq. (2.8). Thus, to a good approximation,

δa_{μ}^{Higgs} ≃ δa_{μ}^{h} ≃ \frac{G_{F} m_{μ}^{2}}{4π^{2}\sqrt{2}} (\frac{m_{μ}^{2}}{m_{h}^{2}} ) tan^{2} β[ ln (\frac{m_{h}^{2}}{m_{μ}^{2}} ) − \frac{7}{6}] .  (2.10)

One can check that a light Higgs boson with a mass of around 10 GeV and with tan β = 35 gives δa_{μ}^{Higgs} ≃ 280 × 10^{−11}, which is within the 90% CL allowed range for δa_{μ}^{NP} quoted in eq. (1.2). Contour lines corresponding to a full numerical evaluation of the Higgs sector one-loop contribution to δa_{μ}^{Higgs} [in units of 10^{−11}] are exhibited in fig. 1, for sin(β − α) = 0 and m_{H} = m_{A} = m_{H^{±}} = 200 GeV.^{6} The relevant experimental bounds are also displayed in fig. 1; these limits are reviewed in section 3. A careful inspection of the excluded region in the m_{h} vs. tan β parameter space shows that a light Higgs boson of around 10 GeV mass and 30 < ∼ tan β < ∼ 35

δa_{μ}^{H^{±}} = a_{μ}^{H^{±}}(1–loop) + \frac{1}{180} (\frac{α}{π})^{2} (\frac{m_{μ}}{m_{H^{±}}} )^{2} + O[(\frac{m_{μ}}{m_{H^{±}}} )^{4} ln (\frac{m_{μ}}{m_{H^{±}}} )] .  (2.9)

5Grifols and Pascual [19] found that for a very light charged Higgs boson, the two-loop contribution to a_{μ} is positive and can be larger in magnitude than the one-loop result given in eq. (2.4):

However, the LEP bound on the charged Higgs mass ref. [20], m_{H^{±}} > 80.5 GeV, implies that both the one and two-loop charged Higgs contribution to δa_{μ}^{NP} are negligible.

6The results are insensitive to the values of the heavy Higgs masses above 100 GeV.

Figure 1: Contours of the predicted one-loop Higgs sector contribution to the muon anomalous magnetic moment, δα_{μ}^{Higgs} (in units of 10^{−11}) in the 2HDM, assuming that sin(β − α) = 0, and m_{H} = m_{A} = m_{H^{±}} = 200 GeV (there is little sensitivity to the heavier Higgs masses). The dashed line contour corresponds to the central value of δa_{μ} ≡ a_{μ}^{exp} − a_{μ}^{SM}, as reported in ref. [1]. The contours lines marked 215 and 637 correspond to 90% CL limits for the contribution of new physics to a_{μ} [eq. (1.2)]. The dark-shaded (red) region is excluded by the CUSB Collaboration search for Υ → hγ at CESR [21]. The lightshaded (yellow) region is excluded at 95% CL by the ALEPH and DELPHI searches for e^{+}e^{−} → bb̄h at LEP [22, 23]. In the small hatched region (green) nestled between the two experimentally excluded shaded regions, above the 215 contour line and centered around m_{h} ≃ 10 GeV, the Higgs sector contribution to δa_{μ}^{NP} lies within the 90% CL allowed range [eq. (1.2)].

is permitted. In this parameter regime, we obtain a value for δa_{μ}^{NP} within the 90% CL allowed range of eq. (1.2).

2.2 Decoupling limit: cos(β − α) = 0

In the decoupling limit, where cos(β − α) ≃ 0 and m_{A} ≫ m_{Z}, the couplings of the light Higgs boson, h, are (nearly) identical to those of the SM Higgs boson. As a result, the LEP SM Higgs mass bound of m_{h} > ∼ 113.5 GeV applies. For cos(β−α) = 0, the H couplings to down-type fermion pairs are enhanced by tan β [see eq. (1.10)].

Thus, the Higgs sector contribution to δa_{μ}^{NP} is given by eq. (2.8), with m_{h} replaced by m_{H}. In the decoupling limit, m_{H} ≃ m_{A} ≃ m_{H^{±}} [the mass differences are of O(m_{Z}^{2}/m_{A})]. Setting cos(β − α) = 0 and m_{H} = m_{A} = m_{H^{±}}, we find

δa_{μ}^{Higgs} ≃ \frac{G_{F} m_{μ}^{2}}{4π^{2}\sqrt{2}} (\frac{m_{μ}^{2}}{m_{A}^{2}} ) tan^{2} β[\frac{1}{2} − (\frac{2m_{μ}^{2}}{m_{A}^{2}} ) ln (\frac{m_{A}^{2}}{m_{μ}^{2}} )] .  (2.11)

The contribution of h is not tan β–enhanced and is thus negligible. In is interesting to note that for values of m_{A} < ∼ m_{h} tan β, the heavier (“decoupled”) Higgs bosons actually dominate in the Higgs sector contribution to δa_{μ}^{NP}.^{7} However, for 100 GeV < m_{A} < 1000 GeV, and 30 < tan β < 100, the Higgs sector contribution to a_{μ} ranges from about 5 × 10^{−12} to 5 × 10^{−14}, which is three to five orders of magnitude below what is needed to explain the BNL measurement of a_{μ}.

3. CESR and LEP constraints on the light Higgs boson

Let us consider the 2HDM in which sin(β −α) = 0, tan β ≫ 1 and m_{h} ∼ O(10 GeV), which are necessary conditions if the Higgs sector is to be the source for δa_{μ}^{NP} in the range given by eq. (1.2). The hAZ coupling is maximal [eq. (1.9)], so we must assume that m_{A} is large enough so that e^{+}e^{−} → hA is not observed at LEP. The tree-level hZZ coupling is absent, which implies that the LEP SM Higgs search based on e^{+}e^{−} → Z → Zh does not impose any significant constraints on m_{h}.^{8} However, there are a number of constraints on light Higgs masses that do not rely on the hZZ coupling. For Higgs bosons with m_{h} < ∼ 5 GeV, the SM Higgs boson was ruled out by a variety of arguments that were summarized in ref. [11]. For 5 GeV < ∼ m_{h} < ∼ 10 GeV, the relevant Higgs boson constraint can be derived from the absence of Higgs production in Υ → hγ.

An experimental search for Υ → hγ by the CUSB Collaboration at CESR [21] found no candidates. The Higgs mass limit obtained from this result depends on the theoretical prediction. In addition to the non-relativistic, tree-level prediction of ref. [26], there are three classes of corrections that have been explored in the literature: O(α_{s}) hard QCD corrections [27, 28], relativistic corrections to the nonrelativistic treatment of the bb̄ bound state [29, 30], and bound state threshold corrections [31]. The theoretical picture that emerges is uncertain. The hard QCD corrections are large and suggest that O(α_{s}^{2}) corrections could be significant. In addition, relativistic effects enter at the same order as the O(α_{s}) corrections; both are of O(v^{2}/c^{2}) and the two must be treated consistently. Finally, ref. [31] argued that strong cancellations can occur among various contributions in the threshold region, leading to an additional suppression in rate of about 14 for m_{h} = 8.5 GeV (and even a larger suppression as m_{h} → M_{Υ}). The application of the theoretical analysis of Γ(Υ → hγ) to the CUSB data suggests that values of m_{h} < ∼ 5—7 GeV can be ruled out at 95% CL, although a precise upper limit cannot be obtained due to the theoretical uncertainties outlined above.

7If we formally take m_{A} → ∞, we recover the Standard Model Higgs contribution to a_{μ}.

8Presumably, radiative corrections would lead to a small effective value for sin(β − α). The LEP Higgs search yields an excluded region in the sin(β − α) vs. m_{h} plane, and implies that for m_{h} ∼ 10 GeV, | sin(β − α)| < ∼ 0.06 is not excluded at 95% CL [24, 25].

The above discussion was relevant for obtaining a limit on the mass of the SM Higgs boson. In the 2HDM considered here, tan β ≫ 1, and the prediction for Γ(Υ → hγ) is enhanced by a factor of tan^{2} β. This factor dwarfs the theoretical uncertainties discussed above except for values of m_{h} very close to M_{Υ}. For values of tan β > ∼ 10, the CUSB data can reliably rule out Higgs masses up to about 8 GeV; the excluded region is indicated by the dark (red) shaded region in fig. 1. But, for Higgs masses above 8 GeV, we see that tan β < ∼ 30 if the Higgs sector contribution to δa_{μ}^{NP} lies in the 90% CL range specified in eq. (1.2). For such large values of tan β, the predicted rate for Υ → hγ is increased by at least three orders of magnitude relative to the SM. This factor dwarfs the theoretical uncertainties discussed above except for values of m_{h} very close to M_{Υ}. Thus, we obtain a lower bound of m_{h} > ∼ M_{Υ}.

A second bound on m_{h} can be derived from the non-observation of Higgs bosons at LEP via the process e^{+}e^{−} → hff̄ (f = b, τ). The cross-section for this process depends on the h Yukawa couplings to down-type fermions. In the 2HDM with sin(β − α) = 0, these Yukawa couplings are enhanced (relative to the corresponding SM value) by tan β. Preliminary analyses by the ALEPH and DELPHI Collaborations at LEP based on the search for e^{+}e^{−} → hff̄ (f = b, τ), where h → τ^{+}τ^{−}, bb̄, find no evidence for light Higgs boson production [22, 23]. Combining the two analyses, we exclude at 95% CL the light-shaded (yellow) region of fig. 1. Note that the lower limit on tan β changes discontinuously at 2m_{B}, where B is the lightest Bmeson [m_{B} = 5.279 GeV]. For Higgs masses that lie in the range 2m_{τ} < ∼ m_{h} < ∼ 2m_{B}, the dominant Higgs decay mode is h → τ^{+}τ^{−}.^{9} In this mass range, the ALEPH limit on tan β is better than the corresponding DELPHI limit. In particular, for M_{Υ} < ∼ m_{h} < ∼ 2m_{B}, the ALEPH excluded region implies that tan β < ∼ 35. For values of m_{h} > 2m_{B}, the Higgs decays primarily into bb̄, and the DELPHI limit (which is more powerful than the ALEPH limit in this mass range) completely excludes the region of parameter space in which the Higgs sector contribution to δa_{μ}^{NP} lies in the 90% CL range specified in eq. (1.2).

One other light Higgs process observable at LEP that is sensitive to the Higgs– fermion Yukawa couplings, even in the absence of the ZZh and W^{+}W^{−}h couplings, is the one-loop process Z → hγ. Both up-type and down-type fermions contribute in the loop, so the decay rate in Model I and Model II differs. Ref. [32] analyzes the implication of this process for the general 2HDM with Model II couplings and shows that the LEP experimental constraints in the m_{h} vs. tan β plane for tan β > 1 are weaker than the ones obtained from e^{+}e^{−} → hff̄ discussed above. In Model I, we can can use the results of ref. [32] simply by interchanging tan β and cot β. For m_{h} ∼ 10 GeV, the LEP experimental constraints imply that tan β < 10. Thus, we have an independent reason to conclude that the Model I 2HDM cannot provide an explanation for the BNL measurement of a_{μ}.

9By assumption, tan β ≫ 1 and the rate for h → cc̄ is suppressed by a factor of cot^{2} β.

Finally, one must check the implications of the precision electroweak data for constraining the Type II 2HDM with a light Higgs boson. This data is known to provide an excellent fit to the Standard Model with one Higgs doublet and m_{h} = 86_{−32}^{+48} GeV [33]. Nevertheless, ref. [34] demonstrates that even with a light Higgs mass below 20 GeV, the CP-conserving Type II 2HDM provides an equally good fit to the precision electroweak data.

4. Final Results and Conclusions

If we combine the experimental bounds on the Higgs mass discussed in section 3, we conclude that a light Higgs boson can be responsible for the observed the 2.6σ deviation of the BNL measurement of the muon anomalous magnetic moment at the 90% CL in the framework of a two-Higgs-doublet model with Model II Higgs-fermion Yukawa couplings only if the model parameters satisfy the following requirements:

m_{Υ} < ∼ M_{h} < ∼ 2m_{B} ,
sin(β − α) ≃ 0 ,
30 < ∼ tan β < ∼ 35 .  (4.1)

In addition, H, A and H^{±} must be sufficiently heavy to satisfy the LEP experimental constraints. In the model specified above, the SM Higgs mass bound applies to H so that m_{H} > ∼ 113.5 GeV. The constraint on m_{A} must at least satisfy m_{A} > ∼ 80 GeV, since Z → HA is not observed,^{10} and m_{H^{±}} > ∼ 80.5 GeV [20].

One noteworthy consequence of m_{h} ∼ 10 GeV is the possibility of mixing between the h and the 0^{++} bb̄ bound states χ_{b0}(1P) and χ_{b0}(2P), as discussed in refs. [18] and [35]. As a result, the decay χ_{b0} → τ^{+}τ^{−} should be prominent. The predicted rate is roughly

\frac{Γ(χ_{b0} → τ^{+}τ^{−})}{Γ(χ_{b0} → hadrons)} ≃ \frac{2.5 × 10^{−7} GeV^{2}}{(m_{χ} − m_{h})^{2}} tan^{4} β ,  (4.2)

which is valid for m_{h} near m_{χ} but separated by a few Higgs widths.^{11} Due to the large tan^{4} β enhancement, the predicted branching ratio for χ_{b0} → τ^{+}τ^{−} can be substantial. Remarkably, the Particle Data Group [36] provides no data on possible decay modes of the χ_{b0} other than the radiative decays, χ_{b0} → Υγ, Υ′γ.

10With further LEP analysis, it should be possible to push the limit on m_{A} higher. The large tan β MSSM Higgs analysis implies that m_{h} + m_{A} < ∼ 180 GeV due to the non-observation of e^{+}e^{−} → hA, although this analysis assumes that m_{h} ∼ m_{A}, which is not applicable in the present model.

11If the two masses are within a Higgs width, then the mixing of the two states will be close to maximal [35], and the corresponding τ^{+}τ^{−} branching ratio of both eigenstates would be close to 100% due to the large tan^{4} β enhancement.

Apart from a careful study of χ_{b0} decays, the 2HDM specified by eq. (4.1) could be confirmed or ruled out by a more complete analysis by the LEP Collaborations of their data in search of e^{+}e^{−} → hff̄ (f = b or τ). We note that the ALEPH and DELPHI exclusion plots used in fig. 1 are based on a preliminary analyses and have not formally appeared in the literature. Without employing these LEP limits, the allowed 2HDM parameter space is significantly larger in which h contributes significantly to δa_{μ}^{NP}. We also note that the results of ref. [37] suggest that the tan β limit could be significantly improved to perhaps tan β > ∼ 5 (for m_{h} ∼ 10 GeV) if a complete analysis were performed using all of the LEP data. The potential significance of such a result should be clear from fig. 1.

In the absence of further information from the LEP collider, one must wait for a further improvement of the BNL measurement of the muon anomalous magnetic moment. A factor of four increase in data is expected when the data sets from the 2000 and 2001 runs are fully analyzed . If the significance of a nonzero result for δa_{μ}^{NP} increases, it will be crucial to discover the source of the new physics. To further constrain the Higgs sector contribution to δa_{μ}^{NP}, a high energy e^{+}e^{−} linear collider that can perform precision studies of Higgs processes is required [38]. One must either discover a light Higgs boson with m_{h} ∼ 10 GeV or improve the present constraints in the m_{h} vs. tan β plane.

A.D. would like to acknowledge financial support from the Network RTN European Program HPRN-CT-2000-0014 “Physics Across the Present Energy Frontier: Probing the Origin of Mass.” H.E.H. would like to thank H.-P. Nilles and H.K. Dreiner for their hospitality during his visit to the Physikalisches Institut der Universität Bonn, where this work was done. H.E.H. is supported in part by a grant from the U.S. Department of Energy.

References

[1] R. Prigl [Muon g − 2 Collaboration], “Status of the BNL Muon g − 2 Experiment,” hep-ex/0101042.

[2] A. Czarnecki and W. J. Marciano, “The muon anomalous magnetic moment: A harbinger for new physics,” hep-ph/0102122, and references therein.

[3] R. Barate et al. [ALEPH Collaboration], “Observation of an excess in the search for the standard model Higgs boson at ALEPH,” Phys. Lett. B 495 (2000) 1 [hepex/0011045]; P. Abreu et al. [DELPHI Collaboration], “Search for the Standard Model Higgs boson at LEP in the year 2000,” Phys. Lett. B 499 (2001) 23 [hep-ex/0102036]; M. Acciarri et al. [L3 Collaboration], “Search for the standard model Higgs boson in e^{+}e^{−} collisions at \sqrt{s} up to 202 GeV,” hep-ex/0012019; G. Abbiendi et al. [OPAL Collaboration], “Search for the standard model Higgs boson in e^{+}e^{−} collisions at \sqrt{s} = 192 GeV—209 GeV,” Phys. Lett. B 499 (2001) 38 [hep-ex/0101014].

[4] M. Krawczyk and J. Zochowski, “Constraining the two Higgs doublet model by present and future (g − 2)_{μ} data,” Phys. Rev. D 55 (1997) 6968 [hep-ph/9608321].

[5] A. Sopczak, “Higgs Physics at LEP-1”, in preparation.

[6] L. Everett, G.L. Kane, S. Rigolin and L. Wang, “Implications of muon g − 2 for supersymmetry and for discovering superpartners directly,” hep-ph/010245.

[7] J.L. Feng and K.T. Matchev, “Supersymmetry and the anomalous anomalous magnetic moment of the muon,” hep-ph/0102146.

[8] E.A. Baltz and P. Gondolo, “Implications of muon anomalous magnetic moment for supersymmetric dark matter,” hep-ph/0102147.

[9] U. Chattopadhyay and P. Nath, “Upper limits on sparticle masses from g − 2 and the possibility for discovery of SUSY at colliders and in dark matter searches,” hepph/0102157.

[10] S. Komine, T. Moroi and M. Yamaguchi, “Recent Result from E821 Experiment on Muon g − 2 and Unconstrained Minimal Supersymemtric Standard Model,” hepph/0102204.

[11] J.F. Gunion, H.E. Haber, G.L. Kane and S. Dawson, The Higgs Hunter’s Guide (Addison-Wesley, Reading, MA, 1990).

[12] S.L. Glashow and S. Weinberg, “Natural Conservation Laws For Neutral Currents,” Phys. Rev. D 15 (1977) 1958; E.A Paschos, “Diagonal Neutral Currents,” Phys. Rev. D 15 (1977) 1966.

[13] L.J. Hall and M.B. Wise, “Flavor Changing Higgs Boson Couplings,” Nucl. Phys. B 187 (1981) 397.

[14] H.E. Haber and Y. Nir, “Multiscalar Models With A High-Energy Scale,” Nucl. Phys. B 335 (1990) 363; H.E. Haber, “Nonminimal Higgs sectors: The Decoupling limit and its phenomenological implications,” hep-ph/9501320, in Proceedings of the US– Polish Workshop, Warsaw, Poland, September 21–24, 1994, edited by P. Nath, T. Taylor, and S. Pokorski (World Scientific, Singapore, 1995) pp. 49–63; H.E. Haber and J.F. Gunion, “The CP-conserving two-Higgs-doublet model and its decoupling limit,” in preparation.

[15] R. Jackiw and S. Weinberg, “Weak Interaction Corrections to the Muon Magnetic Moment and to Muonic Atom Energy Levels,” Phys. Rev. D 5 (1972) 2396.

[16] K. Fujikawa, B.W. Lee and A.I. Sanda, “Generalized Renormalizable Gauge Formulation of Spontaneously Broken Gauge Theories,” Phys. Rev. D 6 (1972) 2923.

[17] J.P. Leveille, “The Second Order Weak Correction to g − 2 of the Muon in Arbitrary Gauge Models,” Nucl. Phys. B 137 (1978) 63.

[18] H.E. Haber, G.L. Kane and T.Sterling, “The Fermion Mass Scale and Possible Effects of Higgs Bosons on Experimental Observables,” Nucl. Phys. B 161 (1979) 493.

[19] J.A. Grifols and R. Pascual, “Contribution of Charged Higgs Bosons to the Anomalous Magnetic Moment of the Muon,” Phys. Rev. D 21 (1980) 2672.

[20] T. Junk [reporting for the LEP Collaborations], “Searches at LEP,” presented at the 5th International Symposisum on Radiative Corrections (RADCOR-2000), Carmel, CA, USA, 11–15, September, 2000, hep-ex/0101015.

[21] P. Franzini et.al [CUSB Collaboration], “Limits On Higgs Bosons, Scalar Quarkonia, and Eta (B)’S From Radiative Upsilon Decays,” Phys. Rev. D 35 (1987) 2883. J. Lee-Franzini, in Proceedings of the XXIV International Conference on High Energy Physics, Munich, Germany, 1988, edited by R. Koffhaus and J.H. Kühn (springer-Verlag, Berlin, 1989) p. 1432.

[22] J.B. de Vivie and P. Janot [ALEPH Collaboration], “Search for a Light Higgs Boson in the Yukawa Process,” PA13-027 contribution to the International Conference on High Energy Physics, Warsaw, Poland, 25–31 July 1996.

[23] J. Kurowska, O. Grajek and P. Zalewski [DELPHI Collaboration], “Search for Yukawa production of a light neutral Higgs at LEP 1,” CERN-OPEN-99-385.

[24] P. Janot, private communication. Previous bounds have been reported in ref. [25].

[25] A. Sopczak, “Status of Higgs hunting at LEP: Five years of progress,” hep-ph/9504300.

[26] F. Wilczek, “Decays of Heavy Vector Mesons into Higgs Particles,” Phys. Rev. Lett. 39 (1977) 1304.

[27] M.I. Vysotsky, “Strong Interaction Corrections to Semiweak Decays: Calculation of the V → Hγ Decay Rate with α_{s} Accuracy,” Phys. Lett. B 97 (1980) 159.

[28] P. Nason, “QCD Radiative Corrections to Upsilon Decay into Scalar Plus Gamma and Pseudoscalar Plus Gamma,” Phys. Lett. B 175 (1986) 223.

[29] S.N. Biswas, A. Goyal and J. Pasupathy, “Radiative Decay of Quarkonium Into Higgs Scalar,” Phys. Rev. D 32 (1985) 1844.

[30] I. Aznauryan, S. Grigoryan, and S. Matinyan, JETP Lett. 43 (1986) 646.

[31] G. Faldt, P. Osland and T.T. Wu, “Relativistic Theory of The Decay of Upsilon Into Higgs + Photon,” Phys. Rev. D 38 (1988) 164.

[32] M. Krawczyk, J. Zochowski and P. Mattig, “Process Z → h(A) + γ in the two Higgs doublet model and the experimental constraints from LEP,” Eur. Phys. J. C 8 (1999) 495 [hep-ph/9811256].

[33] J. Erler, “Fundamental parameters from precision tests,” hep-ph/0102143.

[34] P.H. Chankowski, M. Krawczyk and J. Zochowski, “Implications of the precision data for very light Higgs boson scenario in 2HDM(II),” Eur. Phys. J. C 11 (1999) 661 [hep-ph/9905436].

[35] J. Ellis, M.K. Gaillard, D.V. Nanopoulos and C.T. Sachrajda, “Is the Mass of the Higgs Boson about 10 GeV?,” Phys. Lett. B 83 (1979) 339.

[36] D.E. Groom et al. [Particle Data Group], “Review of particle physics,” Eur. Phys. J. C 15 (2000) 1.

[37] J. Kalinowski and M. Krawczyk, “Two-Higgs-doublet models and the Yukawa process at LEP1,” Acta Phys. Polon. B 27 (1996) 961 [hep-ph/9602292].

[38] M. Krawczyk, P. Mattig and J. Zochowski, “The light Higgs window in the 2HDM at GigaZ,” hep-ph/0009201.